Abstract

Many single-mode fiber components include some form of optics, such as lenses or mirrors, for collecting light from a source fiber or laser and concentrating it on a receiving fiber. For such components there is a direct and simple relationship between coupling efficiency and optical aberrations. This paper combines fiber-coupling fundamentals, classical optics, and diffraction theory to provide a compact description of coupling efficiency that includes the effects of aberrations, fiber misalignments, and fiber-mode mismatch.

© 1982 Optical Society of America

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References

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  1. M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, 1975).
  2. W. T. Welford, Aberrations of the Symmetrical Optical System (Academic, London, 1974).
  3. D. P. Felder, J. Opt. Soc. Am. 47, 913 (1957).
    [CrossRef]
  4. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, San Francisco, 1968).
  5. J. D. Gaskill, Linear Systems, Fourier Transforms, and Optics (Wiley, New York, 1978).
  6. K. Tatsuno, A. Arimoto, Appl. Opt. 20, 3520 (1981).
    [CrossRef] [PubMed]
  7. D. Marcuse, Bell Syst. Tech. J. 56, 703 (1977).
  8. R. H. Stolen, Appl. Opt. 14, 1533 (1975).
    [CrossRef] [PubMed]
  9. H. Kogelnik, T. Li, Proc. IEEE 54, 1312 (1966).
    [CrossRef]
  10. F. Oberhettinger, L. Badii, Tables of Laplace Transforms (Springer, New York, 1973), p. 43.
  11. M. Abramowitz, I. A. Stegan, Eds., Handbook of Mathematical Functions (Dover, New York, 1975), p. 360, Eq. 9.1.21.
  12. K. B. Paxton, W. Streifer, Appl. Opt. 10, 2090 (1971).
    [CrossRef] [PubMed]
  13. A. Papoulis, Probability, Random Variables, and Stochastic Processes (McGraw-Hill, New York, 1965), p. 159, Sec. 5.5.

1981 (1)

1977 (1)

D. Marcuse, Bell Syst. Tech. J. 56, 703 (1977).

1975 (1)

1971 (1)

1966 (1)

H. Kogelnik, T. Li, Proc. IEEE 54, 1312 (1966).
[CrossRef]

1957 (1)

Arimoto, A.

Badii, L.

F. Oberhettinger, L. Badii, Tables of Laplace Transforms (Springer, New York, 1973), p. 43.

Born, M.

M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, 1975).

Felder, D. P.

Gaskill, J. D.

J. D. Gaskill, Linear Systems, Fourier Transforms, and Optics (Wiley, New York, 1978).

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, San Francisco, 1968).

Kogelnik, H.

H. Kogelnik, T. Li, Proc. IEEE 54, 1312 (1966).
[CrossRef]

Li, T.

H. Kogelnik, T. Li, Proc. IEEE 54, 1312 (1966).
[CrossRef]

Marcuse, D.

D. Marcuse, Bell Syst. Tech. J. 56, 703 (1977).

Oberhettinger, F.

F. Oberhettinger, L. Badii, Tables of Laplace Transforms (Springer, New York, 1973), p. 43.

Papoulis, A.

A. Papoulis, Probability, Random Variables, and Stochastic Processes (McGraw-Hill, New York, 1965), p. 159, Sec. 5.5.

Paxton, K. B.

Stolen, R. H.

Streifer, W.

Tatsuno, K.

Welford, W. T.

W. T. Welford, Aberrations of the Symmetrical Optical System (Academic, London, 1974).

Wolf, E.

M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, 1975).

Appl. Opt. (3)

Bell Syst. Tech. J. (1)

D. Marcuse, Bell Syst. Tech. J. 56, 703 (1977).

J. Opt. Soc. Am. (1)

Proc. IEEE (1)

H. Kogelnik, T. Li, Proc. IEEE 54, 1312 (1966).
[CrossRef]

Other (7)

F. Oberhettinger, L. Badii, Tables of Laplace Transforms (Springer, New York, 1973), p. 43.

M. Abramowitz, I. A. Stegan, Eds., Handbook of Mathematical Functions (Dover, New York, 1975), p. 360, Eq. 9.1.21.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, San Francisco, 1968).

J. D. Gaskill, Linear Systems, Fourier Transforms, and Optics (Wiley, New York, 1978).

M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, 1975).

W. T. Welford, Aberrations of the Symmetrical Optical System (Academic, London, 1974).

A. Papoulis, Probability, Random Variables, and Stochastic Processes (McGraw-Hill, New York, 1965), p. 159, Sec. 5.5.

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Figures (20)

Fig. 1
Fig. 1

Coupling from a source fiber (or laser) to a receiving fiber through a general optical system. The receiving fiber lies near an image of the source fiber.

Fig. 2
Fig. 2

Overview of the geometry involved in fiber-to-fiber (or laser-to-fiber) coupling through an optical imaging system. Four planes are important: the source and image planes containing the two fiber endfaces, and the entrance- and exit-pupil planes of the optical system.

Fig. 3
Fig. 3

Details of the geometry for analysis of the coupling-efficiency problem illustrating the first-order properties of the optical system. Included are the source-fiber position S, its image position S′, and the nearby receiving-fiber position I. Also shown are the source plane (s), the entrance-pupil plane (e), the exit-pupil plane (e′), the receiving-fiber image plane (i), and appropriate points and coordinates in those planes.

Fig. 4
Fig. 4

Geometry for the fiber-coupling analysis, illustrating the source-fiber field distributions that occur in the entrance-pupil (e) and exit-pupil (e′) planes. These distributions may be de-centered by the amounts X ¯ o s and X ¯ o s if the source-fiber axis is not directed toward the center of the entrance pupil.

Fig. 5
Fig. 5

Image-space geometry for the coupling-efficiency problem illustrating an ideal reference wave-front ∑ that converges toward the receiving fiber (point I) and the actual wave-front ∑′ emerging from the exit pupil. The difference between the two is called the wave-front aberration W ( X ¯ e ). Such aberrations cause a reduction in coupling efficiency because they represent an improper phase between the two distributions to be coupled.

Fig. 6
Fig. 6

Simple optical system, where two single-mode fibers are coupled by a single-element thin lens. The source fiber lies on the optic axis and the stop, entrance pupil, and exit pupil all coincide and lie at the thin lens itself. Only fiber misalignment and spherical aberration contribute to coupling losses.

Fig. 7
Fig. 7

Aperture size effects on coupling efficiency. The curves are for a lens with no aberrations and for Gaussian source- and receiving-fiber distributions with a size mismatch σ = h s / h r ; taken from Eq. (11)

Fig. 8
Fig. 8

Linear fiber misalignment effects on coupling efficiency. Curves in (a) are for longitudinal misalignment W 20 = ( W 020 + W 220 η 2 ) ρ s 2. Curves in (b) are for lateral misalignment W 11 = W 1 x 2 + W 1 y 2. In both (a) and (b) the size mismatch between Gaussian source- and receiving-fiber distributions is σ = h s / h r ; taken from Eq. (14).

Fig. 9
Fig. 9

Angular fiber misalignment effects on coupling efficiency. The curves are for a lens with no aberrations for Gaussian source- and receiving-fiber distributions with a size mismatch σ = h s / h r and for an exit-pupil shift Δ X / h r = σ x 0 2 + y 0 2; taken from Eq. (18).

Fig. 10
Fig. 10

Wave-front aberration functions illustrating how spherical aberration can be partially balanced by a focal shift. The wave front in (a) shows spherical aberration only (W40), and that in (b) shows the same amount of spherical aberration balanced by a defocus of W20 = −W40.

Fig. 11
Fig. 11

Spherical aberration and focal shift effects on coupling efficiency for W 40 = R 040 ρ s 4 and W 20 = ( W 020 + W 220 η 2 ) ρ s 2. The family of curves illustrates that for each level of spherical aberration (W40) there is a unique focal shift (W20) that maximizes the coupling efficiency. For all curves the Gaussian source- and receiving-fiber distributions are the same size (i.e., σ = 1); taken from Eq. (23).

Fig. 12
Fig. 12

Optical system with the source-fiber off axis, illustrating a case where the third-order aberration terms for coma, astigmatism, field curvature, and distortion must be considered.

Fig. 13
Fig. 13

Coma and lateral shift effects on coupling efficiency, illustrating that coma can be partially balanced by lateral shifts (δy) of the receiving fiber. For all curves the Gaussian source- and receiving-fiber distributions are the same size (i.e., σ = 1) and the aberration coefficients are given by W 31 = W 131 η ρ s 3 and W1y = (W01y + W311η3)ρs. Orthogonal lateral shifts (δx) do not balance coma but produce fiber misalignment and reduced coupling efficiency; taken from Eq. (26).

Fig. 14
Fig. 14

Practical lens configurations in which astigmatism dominates all other third-order aberrations. In configuration (a) two identical doublets, each perfectly corrected for spherical aberration, are arranged symmetrically. In configuration (b) a ½ pitch GRIN-rod lens with a sech(r/r0) index profile couples the two fibers.12 In both cases the magnification is −1, there is no spherical aberration, and the systems are symmetrical about a point located midway between the two fibers, so coma and distortion are absent.

Fig. 15
Fig. 15

Astigmatism and focal shift effects on coupling efficiency, illustrating that astigmatism can be partially balanced by focal shifts of the receiving fiber. For all curves, the Gaussian source- and receiving-fiber distributions are the same size (i.e., σ = 1), and the aberration coefficients are given by W 22 = W 222 η 2 ρ s 2 and W 20 = ( W 020 + W 220 η 2 ) ρ s 2; taken from Eq. (29).

Fig. 16
Fig. 16

A fiber demultiplexer with multiple-receiving fibers illustrating the effects of field curvature. The sketch in (a) shows the source-fiber on axis and an angularly dispersive element producing multiple images at the receiving fibers. The ends of the three receiving fibers lie in a common plane. The two ¼ pitch GRIN-rod lenses are assumed to have a parabolic index profile so there is no astigmatism.12 Spherical aberration (W40) and defocus (W20) produce the dominant effects for this configuration. The curve in (b) shows the effect of field curvature on coupling efficiency if the receiving fibers are equally spaced by η′ and the central fiber is aligned for best coupling. The curve in (c) illustrates the same effect if the central fiber is misfocused to improve the coupling efficiency of the outer two fibers. The curves for both (b) and (c) are taken from Fig. 11 with W40/λ = 0.1 and W 20 = ( W 020 + W 220 η 2 ) ρ s 2.

Fig. 17
Fig. 17

Effect of random wave-front variations on coupling efficiency. This curve indicates the reduction in coupling efficiency that can be applied to a lens with any amount of aberration. The Gaussian source- and receiving-fiber distributions are the same size (i.e., σ = 1) and Wpp = 4σw, where σw is the rms wave-front perturbation; taken from Eq. (35).

Fig. 18
Fig. 18

Coupling efficiency for the third-order aberrations, at best alignment conditions, and for random wave-front perturbations, all illustrated on a common aberration scale. In general, the coupling efficiency is 80% (1-dB loss) for an aberration of ~0.3 waves, regardless of the type of aberration.

Fig. 19
Fig. 19

Geometry for the scalar diffraction theory analysis as described in Appendix A. The lens is represented only by the entrance- and exit-pupil planes e and e′ by the image and pupil magnifications m and me, and by the aberrations WL in the exit pupil that result from a spherical wave incident on the entrance pupil.

Fig. 20
Fig. 20

Image-plane geometry, defining linear fiber misalignments δ η ¯ i and δR′, and emphasizing the differences between η ¯ and η ¯ i and between δ η ¯ and δ η ¯ i.

Equations (66)

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T = Ψ S L Ψ R d a 2 .
W ( ρ , ϕ ) = ( W 020 ρ 2 ) Focal shift ( δ R ) + W 220 η 2 ρ 2 Field curvature + W 040 ρ 4 Spherical aberration + W 222 η 2 ρ 2 cos 2 ϕ Astigmatism + ( W 01 x ρ sin ϕ ) Lateral shift ( δ x ) + ( W 01 y ρ cos ϕ ) Lateral shift ( δ y ) + W 311 η 3 ρ cos ϕ Distortion + W 131 η ρ 3 cos ϕ Coma + .
W ( X ¯ e / h s ) = W 20 r 2 + W 40 r 4 + W 1 y r cos ϕ + W 31 r 3 cos ϕ + W 1 x r sin ϕ + W 22 r 2 cos 2 ϕ + ,
W 1 x = W 01 x ρ s , W 1 y = ( W 01 y + W 311 η 3 ) ρ s , W 20 = ( W 020 + W 220 η 2 ) ρ s 2 , W 40 = W 040 ρ s 4 , W 31 = W 131 η ρ s 3 , W 22 = W 222 η 2 ρ s 2 ,
T = Ψ S ( X ¯ e - X ¯ o s ) exp [ - i k W ( X ¯ e ) ] Ψ R ( X ¯ e - X ¯ o r ) d X ¯ e 2 ,
Ψ S ( X ¯ e ) = m λ R ψ S ( X ¯ s ) exp ( - i k m R X ¯ s · X ¯ e ) d X ¯ s , ] Ψ R ( X ¯ e ) = 1 λ R i ψ R ( X ¯ i ) exp ( - i k R i X ¯ i · X ¯ e ) d X ¯ i .
T = Ψ S ( α ¯ ) exp [ - i k W ( α ¯ ) Ψ R ( α ¯ ) d α ¯ 2 ,
Ψ S ( α ¯ ) = m λ ψ S ( X ¯ s ) exp ( - k m α ¯ · X ¯ s ) d X ¯ s , Ψ R ( α ¯ ) = 1 λ ψ R ( X ¯ i ) exp ( - i k α ¯ · X ¯ i ) d X ¯ i .
Ψ S ( X ¯ e ) = 2 π 1 h s exp ( - X ¯ e · X ¯ e h s 2 ) , Ψ R ( X ¯ e ) = 2 π 1 h r exp ( - X ¯ e · X ¯ e h r 2 ) ,
T = | 2 σ π exp [ - ( 1 + σ 2 ) x ^ · x ^ ] exp [ - i 2 π W ( x ^ ) λ ] d x ^ | 2 ,
T = | 2 σ π 0 r max exp [ - ( 1 + σ 2 ) r 2 ] 2 π r d r | 2 .
T = ( σ β ) 2 [ 1 - exp ( - 2 β r max 2 ) ] 2 ,
W ( x ^ ) = W 1 x x + W 1 y y + W 20 ( x 2 + y 2 ) ,
W 1 x = δ x h s R             W 1 y = δ y h s R             W 20 = δ R h s 2 R 2 .
T = ( 2 σ π ) 2 | - exp { - [ ( 1 + σ 2 ) + i 2 π × ( W 20 λ ) ] x 2 } exp [ - i 2 π ( W 1 x λ ) x ] d x | 2 × a corresponding y integral 2 .
T = A exp { - B [ ( π W 1 x λ ) 2 + ( π W 1 y λ ) 2 ] } ,
A = σ 2 β 2 + ( π W 20 / λ ) 2             B = β β 2 + ( π W 20 / λ ) 2 .
x 0 = R sin δ θ x h s             y 0 = R sin δ θ y h s .
Ψ R ( x ^ ) = 2 π 1 h r exp { - σ 2 [ ( x - x 0 ) 2 + ( y - y 0 ) 2 ] } .
T = ( 2 σ π ) 2 | exp ( - σ 2 x 0 2 ) - exp [ - ( 1 + σ 2 ) x 2 ] exp ( 2 σ 2 x 0 x ) d x | 2 × a corresponding y integral 2 ,
T = A exp [ - C ( x 0 2 + y 0 2 ) ] ,
T = A exp { - B [ ( π W 1 x / λ ) 2 + ( π W 1 y / λ ) 2 ] } exp [ - C ( x 0 2 + y 0 2 ) ] × exp { - D [ x 0 ( π W 1 x / λ ) + y 0 ( π W 1 y / λ ) ] } ,
A = σ 2 β 2 + ( π W 20 / λ ) 2 , B = β β 2 + ( π W 20 / λ ) 2 , C = σ 2 [ β + 2 ( π W 20 / λ ) ] β 2 + ( π W 20 / λ ) 2 , D = 2 σ 2 ( π W 20 / λ ) β 2 + ( π W 20 / λ ) 2 .
W ( x ^ ) = W 40 r 4 + W 20 r 2 .
T = | 2 σ π 0 exp [ - ( 1 + σ 2 ) r 2 ] × exp [ - i 2 π ( W 40 λ r 4 + W 20 λ r 2 ) ] 2 π r d r | 2 .
T = | 2 σ 0 exp [ - 2 ( β + i π W 20 λ ) y ] exp ( - i 2 π W 40 λ y 2 ) d y | 2 ,
T = π 2 · σ 2 ( π W 40 / λ ) exp ( Z 2 ) erfc ( Z ) 2 ,
Z = 1 2 · 1 π W 40 λ [ ( β + π W 20 λ ) - i ( β - π W 20 λ ) ] .
W ( x ^ ) = W 31 r 3 cos ϕ + W 1 y r cos ϕ .
T = | 2 σ π 0 exp ( - 2 β r 2 ) 0 2 π exp [ ( - i 2 π ) × ( W 31 λ r 3 + W 1 y λ r ) cos ϕ ] d ϕ r d r | 2 .
T = | 2 σ π 0 exp ( - 2 β r 2 ) 2 π J 0 [ 2 π ( W 31 λ r 3 + W 1 y λ r ) ] r d r | 2 .
W ( x ^ ) = W 22 r 2 cos 2 ϕ + W 20 r 2 ,
T = | 2 σ π 0 exp ( - 2 β r 2 ) exp ( - i 2 π W 20 λ r 2 ) × 0 2 π exp ( - i 2 π W 22 λ r 2 cos 2 ϕ ) d ϕ r d r | 2 .
T = | 2 π π 0 exp ( - 2 β r 2 ) exp [ - i 2 π ( W 20 λ + 1 2 W 22 λ ) r 2 ] × [ 2 π J 0 ( 2 π · 1 2 W 22 λ r 2 ) ] r d r | 2 .
C 0 i i = Ψ S exp [ - i k W T ( X ¯ e ) ] i Ψ R d X ¯ e .
C 0 i i = exp [ - i k W R ( X ¯ e ) ] i Ψ S exp [ - i k W ( X ¯ e ) ] Ψ R d X ¯ e .
C 0 i i = exp [ - i k W R ( X ¯ e ) ] i C 0 ,
exp [ - i k W R ( X ¯ e ) ] i = exp [ - k 2 2 { W r 2 ( X ¯ e ) - W R ( X ¯ e ) 2 } ] .
exp [ - i k W R ( X ¯ e ) ] i = exp [ - 2 π 2 ( σ w λ ) 2 ] ,
T R = exp [ - ( 2 π · σ w λ ) 2 ] T ,
Ψ S ( X ¯ e ) = 1 λ R Ψ S ( X ¯ ) exp ( - i k R X ¯ · X ¯ e ) d X ¯ ,
exp ( i k 2 R X ¯ e · X ¯ e ) .
A s ( X ¯ s ) = Ψ S ( X ¯ s - η ¯ ) exp [ i k ( X ¯ s - η ¯ ) · f ^ S ] .
A e ( X ¯ e ) = exp ( i k r ) i λ R A s ( X ¯ s ) exp [ - i k R ( X ¯ s - η ¯ ) · ( X ¯ e - η ¯ ) ] d X ¯ s ,
r R - η ¯ · X ¯ e R + X ¯ e · X ¯ e 2 R , R k 2 r s 2 .
A e ( X ¯ e ) = exp ( i k r ) i λ R ψ s ( X ¯ ) exp [ - i k R X ¯ · ( X ¯ e - X ¯ o s ) ] d X ¯ ,
A e ( X ¯ e ) = exp ( i k r ) i Ψ S ( X ¯ e - X ¯ o s ) .
A e ( X ¯ e ) = exp ( - i k r ) i Ψ S ( X ¯ e - X ¯ o s ) exp [ - i k W L ( X ¯ e ) ] ,
Ψ S ( X ¯ e ) = 1 m e Ψ S ( X ¯ e m e ) , X ¯ o s = m e X ¯ o s , r R - η ¯ · X ¯ e R + X ¯ e · X ¯ e 2 R .
Ψ S ( X ¯ e ) = 1 m e λ R ψ S ( X ¯ ) exp ( - i k m e R X ¯ · X ¯ e ) d X ¯ .
Ψ S ( X ¯ e ) = m λ R ψ S ( X ¯ ) exp ( - i k m R X ¯ · X ¯ e ) d X ¯ .
A i ( X ¯ i ) = 1 i λ R i A e ( X ¯ e ) exp ( i k r i ) d X ¯ e ,
r i R i + ( X ¯ e - X ¯ i ) · ( X ¯ e - X ¯ i ) - η ¯ i · η ¯ i 2 R i .
X ¯ i = η ¯ i + X ¯ ,             η ¯ i = η ¯ + δ η ¯ ,             R i R + δ R .
r i r + δ R - W M ( X ¯ e ) + X ¯ · X ¯ 2 R i - X ¯ · ( X ¯ e - η ¯ i ) R i ,
W M ( X ¯ e ) = ( R δ η ¯ - δ R η ¯ ) · X ¯ e R 2 + δ R X ¯ e · X ¯ e 2 R 2 .
δ η ¯ i = δ η ¯ - δ R ( η ¯ R ) ,
W M ( X ¯ e ) = δ η ¯ i · X ¯ e R + δ R X ¯ e · X ¯ e 2 R 2 .
A i ( X ¯ i ) = exp ( i k δ R ) · exp ( i k 2 R i X ¯ · X ¯ ) i 2 λ R i Ψ S ( X ¯ e - X ¯ o s ) × exp [ - i k W ( X ¯ e ) ] × exp [ - i k R i X ¯ · ( X ¯ e - η ¯ i ) ] d X ¯ e ,
W ( X ¯ e ) = W L ( X ¯ e ) + W M ( X ¯ e ) .
A f ( X ¯ ) = A i ( X ¯ i ) exp [ i k ( X ¯ i - η ¯ i ) · f ^ R ] ,
C 0 = A f ( X ¯ ) ψ R ( X ¯ ) d X ¯ ,
C 0 = - exp ( i k δ R ) λ R i Ψ S ( X ¯ e - X ¯ o s ) exp [ - i k W ( X ¯ e ) ] × ψ R ( X ¯ ) exp [ - i k R i X ¯ · ( X ¯ e - η ¯ i ) ] exp ( i k X ¯ · f ^ R ) d X ¯ d X ¯ e ,
Ψ R ( X ¯ e ) = 1 λ R i ψ R ( X ¯ ) exp ( - i k R i X ¯ e · X ¯ ) d X ¯ .
C 0 = - exp ( i k δ R ) Ψ S ( X ¯ e - X ¯ o s ) × exp [ - i k W ( X ¯ e ) ] Ψ R ( X ¯ e - X ¯ o r ) d X ¯ e ,
R k 2 r s 2 ,             R i k 2 r r 2 ,

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